A shell that moves in a ballistic path, see FIG. 2, will rotate the speed vector around an axis that lies in a horizontal plane. The rotation of the speed vector will take place around a plane-fixed y-axis yPF. The plane-fixed coordinate system is defined in such a way that its origin follows the centre of gravity of the shell. The plane-fixed x-axis points forward in the shell along the axis of symmetry. The plane-fixed y-axis points to the right, viewed from the back, and lies in a plane that has the g-vector (g=gravitation) as a perpendicular. Finally, the plane-fixed z-axis points in such a way that the coordinate system has a right-hand rotation.
When rotation sensors are mounted in the shell, it is convenient to define a body-fixed coordinate system by the designation BF (Body Fixed), see FIG. 1. When the shell rotates around the axis of symmetry, an angle arises between the y-axis and z-axis of the plane-fixed coordinate system and the respective y-axis and z-axis of the body-fixed coordinate system. This angle is designated “φ” in FIG. 1 and is called in the following the roll angle.
If three rotation-measuring sensors are mounted in the shell in such a way that they measure the rotation around respective body-fixed coordinate axes directly or via a linear combination, the inertial rotation vector can be expressed in the rotational directions ωxBF, ωyBF, ωzBF of the body-fixed coordinate system.
The rotation around the plane-fixed y-axis can then be expressed as measurement signals from the body-fixed rotation sensor signals and the roll angle can thereafter be calculated.
                              ω          yBF                =                ⁢                              ω            yPF                    ·                      cos            ⁡                          (              ϕ              )                                                                        ω          zBF                =                ⁢                              -                          ω              yPF                                ·                      sin            ⁡                          (              ϕ              )                                                              ϕ        =                ⁢                  a          ⁢                                          ⁢                      tan            ⁡                          (                                                -                                      ω                    zBF                                                                    ω                  yBF                                            )                                          
However, the shell is acted upon not only by the g-vector but also by the atmosphere and, in particular, by wind turbulence in the atmosphere. This gives rise to moment interferences around the coordinate axes yBF and zBF. This, in turn, gives rise to rotations in ωyBF and ωzBF. These rotations can be greater by the power of 10 than the rotation ωyPF caused by the effect of the g-vector on the path. In practice, therefore, the simple formula above can not be used to calculate the roll angle directly. In order to handle the body-fixed rotation sensor signals, the signals are therefore filtered. It has, however, proved difficult to filter effectively measurement signals that are non-linear. For example, linear filters of the Kalman type have proved to be difficult to use.